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An oscillation criterion in \(4{th}\)-order neutral differential equations with a continuously distributed delay
Advances in Difference Equations volume 2019, Article number: 336 (2019)
Abstract
In this paper, a class of \(4{th}\)-order neutral delay differential equations with continuously distributed delay is studied. We establish a new oscillation criterion using the Riccati transformation. An example illustrating the results is also given.
1 Introduction
In this work, we consider a \(4{th}\)-order neutral differential equation with a continuously distributed delay of the form
We assume that the following conditions hold:
- \(( H_{1} ) \) :
-
α is a quotient of odd positive integers;
- \(( H_{2} ) \) :
-
p, q, τ, \(g\in C ( [ t_{0},\infty ) ,R ) \), \(r ( t ) \), and \(q(t,\xi )\) are positive, \(0\leq p ( t ) \leq p<1\), \(r ( t ) \in C^{{1}} ( [ t_{0},+\infty ) ) \), \(r^{\prime } ( t ) \geq 0\), \(\tau ( t ) \leq t\), \(g(t,\xi )\leq t\), \(\lim_{t\rightarrow \infty } \tau ( t ) =\infty \), \(\lim_{t\rightarrow \infty }g(t, \xi )=\infty , q(t,\xi )\) is not zero on any half line \([t_{\lambda },\infty )\times {}[ a,b]\), \(t_{\lambda }\geq t_{0}\), for \(t\geq t_{0}\), \(\xi \in {}[ a,b]\), \(g(t,\xi )\) is nondecreasing with respect to ξ.
- \(( H_{3} ) \) :
-
There exists a constant \(k>0\) such that \(f ( u ) /u^{{\gamma }}\geq k\) for \(u\neq 0\).
We define the corresponding function \(z ( t ) \) of a solution \(x ( t ) \) of (1) by \(z ( t ) =x ( t ) +p ( t ) x ( \tau ( t ) ) \), we mean a non-trivial real function \(x(t)\in C ( [t _{x},\infty ) ) \), \(t_{x}\geq t_{0}\), satisfying (1) on \([t_{x},\infty )\) and, moreover, having the properties: \(z(t)\), \(z^{\prime }(t)\), \(z^{\prime \prime }(t)\) and \(r ( t ) [ z^{\prime \prime \prime } ( t ) ] ^{\alpha }\) are continuously differentiable for all \(t\in {}[ t_{x}, \infty )\). We consider only those solutions \(x ( t ) \) of (1) which satisfy \(\sup \{ \vert x(t) \vert :t \geq T\}>0\) for any \(T\geq t_{x}\). AÂ solution of (1) is called oscillatory if it has arbitrary large zeros, otherwise it is called nonoscillatory.
The oscillations of higher- and fourth-order differential equations have been studied by several authors, and several techniques have been proposed for obtaining oscillatory criteria for higher- and fourth-order differential equations. For treatments on this subject, we refer the reader to the texts [2, 5, 16,17,18, 21] and the articles [1, 3,4,5,6,7,8,9,10,11,12,13,14,15, 19,20,21,22,23,24,25,26]. In what follows, we review some results that have provided the background and motivation for the present work.
Cesarano and Bazighifan [8], Moaaz et al. [21], and Zhang et al. [26] studied the oscillation of the fourth-order nonlinear differential equation with a continuously distributed delay
Li et al. [19] studied the oscillatory behavior of the fourth-order nonlinear differential equation
Parhi and Tripathy [23] have considered the fourth-order neutral differential equations of the form
and
and established the oscillation and asymptotic behavior of the above equations under the conditions
and
respectively.
Our aim in the present paper is to use the Riccati method to establish new conditions for the oscillation of all solutions of (1) under condition (7).
The proof of our main results is essentially based on the following lemmas.
Lemma 1.1
Let \(\beta \geq 1 \) be a ratio of two numbers, where U and V are constants. Then
Lemma 1.2
If the function z satisfies \(z^{(i)}>0\), \(i=0,1,\ldots,n\), and \(z^{ ( n+1 ) }<0\), then
Lemma 1.3
Let \(h\in C^{n} ( [ t_{0},\infty ) , ( 0,\infty ) ) \). Assume that \(h^{ ( n ) } ( t ) \) is of a fixed sign and not identically zero on \([ t_{0},\infty ) \) and that there exists \(t_{1}\geq t _{0}\) such that \(h^{ ( n-1 ) } ( t ) h^{ ( n ) } ( t ) \leq 0\) for all \(t\geq t_{1}\). If \(\lim_{t\rightarrow \infty }h ( t ) \neq 0\), then for every \(\lambda \in ( 0,1 ) \) there exists \(t_{\lambda }\geq t _{0}\) such that
2 Main results
In this section, we shall establish some oscillation criteria for equation (1).
For convenience, we denote
and
Theorem 2.1
Assume that (7) holds. If there exist positive functions \(\rho ,\vartheta \in C ( [ t_{0},\infty ) ) \) such that
for some \(\mu \in ( 0,1 ) \), where
and either
or
or
then all solutions of (1) are oscillatory.
Proof
Let x be a nonoscillatory solution of equation (1) defined in the interval \([ t_{0},\infty ) \). Without loss of generality, we can assume that \(x ( t ) \) is eventually positive. It follows from (1) that there are two possible cases for \(t\geq t_{1}\), where \(t_{1}\geq t_{0} \) is sufficiently large:
- \(( C_{1} )\) :
-
\(z^{\prime } ( t ) >0\), \(z ^{\prime \prime } ( t ) >0\), \(z^{\prime \prime \prime } ( t ) >0\), \(\bigl( r ( t ) \bigl( z^{\prime \prime \prime } ( t ) \bigr) ^{\alpha } \bigr) <0\),
- \(( C_{2} )\) :
-
\(z^{\prime } ( t ) >0\), \(z^{ \prime \prime } ( t ) <0\), \(z^{\prime \prime \prime } ( t ) >0\), \(\bigl( r ( t ) \bigl( z^{\prime \prime \prime } ( t ) \bigr) ^{\alpha } \bigr) <0\).
Assume that Case \(( C_{1} ) \) holds. Since \(\tau (t) \leq t\) and \(z^{\prime } ( t ) >0\), we get
From equation (1), we see that
so that
Since \(g(t,\xi )\) is nondecreasing with respect to ξ and \(z^{\prime } ( t ) >0\), we have
Thus
Now, we define a generalized Riccati substitution by
Then \(\omega ( t ) >0\). Differentiating and using (19), we obtain
From Lemma 1.2, we have that \(z ( t ) \geq \frac{t}{3}z^{\prime } ( t ) \), and hence
Since \(r^{\prime } ( t ) >0\) and \(( r ( t ) ( z^{\prime \prime \prime } ( t ) ) ^{\alpha } ) ^{\prime }\leq 0\), we get \(z^{(4)} ( t ) <0\). It follows from Lemma 1.3 that
for all \(\mu \in ( 0,1 ) \) and every sufficiently large t. Thus, by (24), (25), and (26), we get
Using Lemma 1.1 with \(U=\frac{\rho ^{\prime } ( t ) }{ \rho ( t ) }\), \(V=\frac{\alpha \mu t^{2}}{2r^{1/\alpha } ( t ) \rho ^{1/\alpha } ( t ) }\) and \(y=\omega \), we get
This implies that
for some \(\mu \in ( 0,1 ) \) which contradicts (12).
Assume that Case \(( C_{2} ) \) holds. Integrating (1) from \(t_{1}\) to t, we obtain
From \(z^{\prime } ( t ) >0\), \(x(t)\geq (1-p(t))z(t) \) and \(g ( s,\xi ) \leq t\), it follows that
From (25), we get
which contradicts (12). Integrating (1) from t to ∞, we obtain
By virtue of \(z^{\prime } ( t ) >0\), \(x(t)\geq (1-p(t))z(t)\), \(g ( s,\xi ) \leq t\), and (25), we obtain
Integrating (33) from \(t_{1}\) to t, we get
This yields
which contradicts (15). Integrating (33) from t to ∞, we get
so that
Now, we define the Riccati substitution
then \(\psi ( t ) >0 \) for \(t\geq t_{1} \) and
From (37) and (38), it follows that
Hence, we have
Integrating (41) from \(t_{1}\) to s, we get
for all large s, which contradicts (16).
The proof of the theorem is complete. □
Let \(\rho ( t ) =t^{3} \) and \(\vartheta ( t ) =t\). As a consequence of Theorem 2.1, we obtain the following oscillation criterion.
Corollary
Assume that (7) holds and for some constant \(\lambda _{0} \in ( 0, 1 ) \)
where
If either (14) or (15) is satisfied, or
then all solutions of (1) are oscillatory.
3 Example
In this section, we give the following example to illustrate our main results.
Example
Consider the differential equation
where \(\nu >0\) is a constant. Let
we get
If we now set \(k=1\), then
and
Thus, by Corollary, every solution of equation (46) is oscillatory.
4 Conclusion
The results of this paper are presented in a form which is essentially new and of high degree of generality. In this paper, using a Riccati transformation technique, we offer some new sufficient conditions which ensure that any solution of Eq. (1) oscillates under the condition \(\int _{t_{0}}^{\infty }\frac{1}{r^{\frac{1}{\alpha }} ( t ) }\,dt=\infty \). Further, we can consider the case of \(z ( t ) =x ( t ) -p ( t ) x ( \tau ( t ) ) \), and we can try to get some oscillation criteria of Eq. (1) in the future work.
References
Agarwal, R., Grace, S., Manojlovic, J.: Oscillation criteria for certain fourth order nonlinear functional differential equations. Math. Comput. Model. 44, 163–187 (2006)
Agarwal, R., Grace, S., O’Regan, D.: Oscillation Theory for Difference and Functional Differential Equations. Kluwer Academic, Dordrecht (2000)
Agarwal, R., Grace, S., O’Regan, D.: Oscillation criteria for certain nth order differential equations with deviating arguments. J. Math. Anal. Appl. 262, 601–622 (2001)
Baculikova, B., Dzurina, J., Graef, J.R.: On the oscillation of higher-order delay differential equations. Math. Slovaca 187, 387–400 (2012)
Bazighifan, O.: Oscillatory behavior of higher-order delay differential equations. Gen. Lett. Math. 2, 105–110 (2017)
Bazighifan, O., Cesarano, C.: Some new oscillation criteria for second-order neutral differential equations with delayed arguments. Mathematics 7, 1–8 (2019)
Bazighifan, O., Elabbasy, E.M., Moaaz, O.: Oscillation of higher-order differential equations with distributed delay. J. Inequal. Appl. 2019, 55 (2019)
Cesarano, C., Bazighifan, O.: Oscillation of fourth-order functional differential equations with distributed delay. Axioms 7, 1–9 (2019)
Cesarano, C., Bazighifan, O.: Qualitative behavior of solutions of second order differential equations. Symmetry 11, 1–8 (2019)
Cesarano, C., Pinelas, S., Al-Showaikh, F., Bazighifan, O.: Asymptotic properties of solutions of fourth-order delay differential equations. Symmetry 11, 1–10 (2019)
Elabbasy, E.M., Hassan, T.S., Moaaz, O.: Oscillation behavior of second order nonlinear neutral differential equations with deviating arguments. Opuscula Mathematica 32, 719–730 (2012)
Grace, S., Graef, J., Tunc, E.: Oscillatory behavior of a third order neutral dynamic equations with distributed delays. Electron. J. Qual. Theory Differ. Equ. 2016, 14 (2016)
Grace, S., Graef, J., Tunc, E.: Oscillatory behavior of second order damped neutral differential equations with distributed deviating arguments. Miskolc Math. Notes 18, 759–769 (2017)
Grace, S., Lalli, B.: Oscillation theorems for nth order nonlinear differential equations with deviating arguments. Proc. Am. Math. Soc. 90, 65–70 (1984)
Graef, J., Tunc, E.: Oscillation of fourth-order nonlinear dynamic equations on time scales. Panam. Math. J. 25, 16–34 (2015)
Gyori, I., Ladas, G.: Oscillation Theory of Delay Differential Equations with Applications. Clarendon, Oxford (1991)
Kiguradze, I., Chanturia, T.: Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations. Kluwer Academic, Drodrcht (1993)
Ladde, G., Lakshmikantham, V., Zhang, B.: Oscillation Theory of Differential Equations with Deviating Arguments. Dekker, New York (1987)
Li, T., Baculikova, B., Dzurina, J., Zhang, C.: Oscillation of fourth order neutral differential equations with p-Laplacian like operators. Bound. Value Probl. 2014, 56 (2014)
Moaaz, O., Chalishajar, D., Bazighifan, O.: Some qualitative behavior of solutions of general class difference equations. Mathematics 7, 1–12 (2019)
Moaaz, O., Elabbasy, E.M., Bazighifan, O.: On the asymptotic behavior of fourth-order functional differential equations. Adv. Differ. Equ. 2017, 261 (2017)
Moaaz, O., Elabbasy, E.M., Muhib, A.: Oscillation criteria for even-order neutral differential equations with distributed deviating arguments. Adv. Differ. Equ. 2019, 297 (2019)
Parhi, N., Tripathy, A.: On oscillatory fourth order linear neutral differential equations-I. Math. Slovaca 54, 389–410 (2004)
Philos, C.: On the existence of nonoscillatory solutions tending to zero at ∞ for differential equations with positive delay. Arch. Math. (Basel) 36, 168–178 (1981)
Tunc, E., Graef, J.: Oscillation results for second order neutral dynamic equations with distributed deviating arguments. Dyn. Syst. Appl. 3, 289–303 (2014)
Zhang, C., Li, T., Saker, S.: Oscillation of fourth-order delay differential equations. J. Math. Sci. 201, 296–308 (2014)
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The authors express their debt of gratitude to the editors and the anonymous referee for accurate reading of the manuscript and beneficial comments.
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Chatzarakis, G.E., Elabbasy, E.M. & Bazighifan, O. An oscillation criterion in \(4{th}\)-order neutral differential equations with a continuously distributed delay. Adv Differ Equ 2019, 336 (2019). https://doi.org/10.1186/s13662-019-2281-3
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DOI: https://doi.org/10.1186/s13662-019-2281-3
MSC
- 34K10
- 34K11
Keywords
- \(4{th}\)-order
- Neutral differential equations
- Oscillatory solutions